Ellipsoidal Vortices Movie (1.3 MB, .avi)
The projects in each of these fields are often computationally intense and would be significantly aided by powerful computational facilities. Some of the work is simulated on massively parallel machines away from our department. On the other hand, a larger portion of the work is often done on lesser workstations thus limiting the scope and effectiveness of our research. HP Itanium2 machines would greatly enhance research capabilities in house. They would open up new avenues and territories in the computational aspect of our research. Below are brief descriptions, pictures and some beautiful and amazing simulations of the research being done here. We can only imagine the immeasurable impact an HP Itanium2 system would have on the quality of our work.
| The quasigeostrophic equations are used to model medium to large scale motions in the ocean and atmospheres. Coherent regions of vorticity are common in the world's oceans and in simulations of these governing equations. Here we use a moment model to describe the evolution of two vortices. This is a system of ordinary differential equations that are simulated numerically. The moments that are computed are then visualized as the surface of an ellipse. |
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 Mixing Movie (13 MB, .avi)
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In this project, the tools of dynamical systems are used to study 3D fluid mixing. The system, which is a limiting case of the navier stokes equations, consists alternatively active rolls about different axes. The most computationally intensive aspect of the movie is the generation of the Lyapunov exponents. These exponents measure the amount of chaos and mixing(dispersion) in the system. That is, high Lyapunov exponents correspond to chaotic behavior (hence, lots of mixing). Small Lyapunov exponents correspond to regular behavior (hence no mixing). It took 17 days on a 2 GHz Pentium 4 to compute all the necessary data for this movie.
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 Vorticity Movie (1.2 MB, .avi) 3D Vorticity Movie (4.5 MB, .mov)
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The figure on the left was created by a Quasi-Geostrophic simulation by Jeff Weiss. It is used to study ocean dynamics, like the eddies that spin off the Gulf Stream, as viewed by a satellite image of surface temperature, right. |
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Mark Petersen, a graduate student who works with Professors Keith Julien
(CU Applied Math) and Jeff Weiss (CU Program in Atmospheric and Oceanic
Science) is writing a numerical model of the Intermediate
Quasi-Geostrophic Potential Vorticity Equation. The Quasi-Geostrophic
equation, first derived by Charney in 1948, has been a very successful
model of mid-latitude fluid flow of the atmosphere and ocean. This new
intermediate model considers cases where rotation is not aligned with
stratification and may have applications for dynamics near the equator.
The numerical model is a three dimensional pseudo-spectral code. Mark is
currently creating a version to run on parallel machines.
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We have been exploring classification and feature extraction methods. A
current area of research has been on Non-negative Matrix Factorization (NMF)
whereby a collection of non-negative data is reduced to a low dimensional
representation consisting of two non-negative factors. Current iterative
factorization techniques are very computationally intensive and we have been
speeding up these techniques using structured initializations. The nature
of this constrained optimization problem is that a basis of local features
and an accompanying weight/coefficient matrix are produced.
In the case of a data set of several hundred facial images, the resulting rank-24 basis contains "parts" of faces, such as eyes, noses, mouths, et cetera. Our approximation then reconstructs the original images (4 are shown here) by computing the (element-by-element) product of these 24 basis elements with the (4) coefficient matrices shown here. |
Computation of Stable and Unstable Manifolds
: Derin Wysham, Graduate Student with Professor Jim Meiss
| Depiction of the stable (blue) and unstable (red) manifolds for a pair of quasi-periodic orbits. These manifolds act as a quick reference as to the types of behavior that are possible for a given physical system. For example, imagine a pendulum acting without friction. The stable and unstable manifolds comprise the dividing line between the action of the pendulum swinging back and forth like a gradfather clock and the action of the pendulum swinging around and around like a softball pitcher's arm. The method used to generate this figure has since been improved. The new scheme can handle higher-dimensional quasi-periodic orbits without the expected enormous increase in computational time. |
 Stable (blue) and Unstable (red) Manifolds
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 Nonlinear Wave Interaction Movie (1.7 MB, .avi)
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The wave movie shows a numerical simulation of a acoustic wave propagating through a two dimensional space with reflecting boundaries. Fast and accurate simulation of propagating acoustic waves are of great importance in for example medical imaging and seismology. The goal of our current research is to develop a fast and highly accurate algorithm for simulation of wave phenomena. In particular, we want to solve so-called inverse problems where a scattered wave is measured in order to reconstruct the internal strucure of an object. |
| We study least-squares finite element methods for linear and nonlinear hyperbolic conservation laws. A large motivation in our research efforts is computational cost. We develop our formulation so that adaptivity in both space and time is achievable. We also focus our efforts on multigrid iterative methods in order to attain efficient solves of the resulting linear systems. |
 This shows the numerical solution to a linear conservation law (pure linear advection) |
 This shows the numerical solution to a nonlinear hyperbolic conservation law (the Burger's equation) |